I do not know how to get the result in the rhs. I know a few things about Taylor Expansions but I do not know how to get the extra term. I would really appreciate if someone could show me a step-by-step method on how to get to the result in the RHS.
Thank you!
The variable is $u=\sqrt{\Delta t}$, and the expression can be re-written as $$\mathrm e^{\bigl(r-\tfrac{\sigma^2}2\bigr)u^2+\sigma u}.$$ Taylor's expansion at order $2$ of $\mathrm e^x$ is $\;1+x+\dfrac{x^2}2+o(x^2)$, so we substitute $\bigl(r-\tfrac{\sigma^2}2\bigr)u^2+\sigma u$ to $x$ in this formula $$\Bigl[\Bigl(r-\frac{\sigma^2}2\Bigr)u^2+\sigma u\Bigr]^2=\Bigl(r-\frac{\sigma^2}2\Bigr)^{\!2}u^4+2\sigma\Bigl(r-\frac{\sigma^2}2\Bigr)u^3+\sigma^2 u^2=\sigma^2 u^2+o(u^2).$$ So we obtain $$1+\Bigl(r-\frac{\sigma^2}2\Bigr)u^2+\sigma u+\frac12\sigma^2 u^2+o(u^2)=1+\sigma u+ru^2+o(u^2).$$